R&DE (Engineers), DRDO. Theories of Failure. Ramadas Chennamsetti

Transcription

1 heories of Failure

2 ummary Maximum rincial stress theory Maximum rincial strain theory Maximum strain energy theory Distortion energy theory Maximum shear stress theory Octahedral stress theory

3 Introduction Failure occurs when material starts exhibiting inelastic behavior Brittle and ductile materials different modes of failures mode of failure deends on loading Ductile materials exhibit yielding lastic deformation before failure ield stress material roerty Brittle materials no yielding sudden failure Factor of safety (F)

4 Introduction Ductile and brittle materials F F ε e ε Ductile material ε.% ε Well defined yield oint in ductile materials F on yielding No yield oint in brittle materials sudden failure F on failure load Brittle material ε 4

5 Introduction tress develoed in the material < yield stress imle axial load x x If x > yielding starts failure ielding is governed by single stress comonent, x imilarly in ure shear only shear stress. If τ max τ > ielding in shear x x Multi-axial stress state?? 5

6 Introduction Various tyes of loads acting at the same time N Axial, moment and torque M Internal ressure and external UDL 6

7 Introduction Multiaxial stress state six stress comonents one reresentative value Define effective / equivalent stress combination of comonents of multiaxial stress state quivalents stress reaching a limiting value roerty of material yielding occurs ield criteria ield criteria define conditions under which yielding occurs ingle yield criteria doesn t cater for all materials election of yield criteria Material yielding deends on rate of loading static & dynamic 7

8 Introduction ield criteria exressed in terms of quantities like stress state, strain state, strain energy etc. ield function > f( ij, ), ij stress state If f( ij, )< > No yielding takes lace no failure of the material If f( ij, ) starts yielding onset of yield If f( ij, ) > -?? ield function develoed by combining stress comonents into a single quantity effective stress > e 8

9 Introduction quivalent stress deends on stress state and yield criteria not a roerty Comare e with yield stress of material ield surface grahical reresentation of yield function, f( ij, ) ield surface is lotted in rincial stress sace Haiagh Westergaard stress sace ield surface closed curve 9

10 Parameters in uniaxial tension Maximum rincial stress Alied stress >,, Maximum shear stress τ max Maximum rincial strain,, ε υ ( )

11 Parameters in uniaxial tension otal strain energy density Distortional energy Linear elastic material U ε Distortional energy [ ] First invariant for deviatoric art > / U U D U V

12 Parameters in uniaxial tension Volumetric strain energy density, U V /K ( ) U U U K K U V D V 6 8 υ ( ) ( ) ( ) G U U U U U D D V D υ υ υ imilarly for ure shear also

13 Failure theories Failure mode Mild steel (M. ) subjected to ure tension M. subjected to ure torsion Cast iron subjected to ure tension Cast iron subjected to ure torsion heories of failure Max. rincial stress theory Rankine Max. rincial strain theory t. Venants Max. strain energy Beltrami Distortional energy von Mises Max. shear stress theory resca Octahedral shear stress theory

14 Max. rincial stress theory Maximum rincial stress reaches tensile yield stress () For a given stress state, calculate rincile stresses,, and ield function f f If, (,, ) max f f > < onset not no of yielding yielding defined 4

15 Max. rincial stress theory ield surface ± >, ± >, ± >, ield surface Reresent six surfaces ield strength same in tension and comression 5

16 Max. rincial stress theory In D case, equations become ± >, ± >, Closed curve tress state inside elastic, outside > ielding - Pure shear test > τ, - τ - For tension > From the above > τ xerimental results ield stress in shear is less than yield stress in tension Predicts well, if all rincial stresses are tensile τ Pure shear 6 τ y τ y

17 Max. rincial strain theory Failure occurs at a oint in a body when the maximum strain at that oint exceeds the value of the maximum strain in a uniaxial test of the material at yield oint yield stress in uniaxial tension, yield strain, ε y / he maximum strain develoed in the body due to external loading should be less than this Princial stresses >, and strains corresonding to these stress > ε, ε and ε 7

18 Max. rincial strain theory trains corresonding to rincial stresses - ε ε ε υ υ υ ( ) ( ) ( ) Maximum of this should be less than ε y For onset of yielding ε > υ ε > υ ε > υ ( ) ( ) ± ± ( ) ± here are six equations each equation reresents a lane 8

19 Max. rincial strain theory ield function f max υ υ i j k For D case i e υ υ j f e k max υ υ i j k i, i, j, k,, j > υ ± > υ ± here are four equations, each equation reresents a straight line in D stress sace k 9

20 Max. rincial strain theory quations υ υ Plotting in stress sace,, υ υ ν y ν ν y ν Failure equivalent stress falls outside yield surface

21 Max. rincial strain theory Biaxial loading For onset of yielding υ ( υ ) ( υ ) Maximum rincial stress theory - Max. rincial strain theory redicts smaller value of stress than max. rincial stress theory Conservative design

22 Max. rincial strain theory Pure shear Princial stresses corresonding to shear yield stress τ y τ y, -τ y For onset of yielding max. rincial strain theory τ y υ τ y τ y ( υ) Relation between yield stress in tension and shear τ y / ( υ) for υ.5 τ y.8 Not suorted by exeriments

23 train energy theory Failure at any oint in a body subjected to a state of stress begins only when the energy density absorbed at that oint is equal to the energy density absorbed by the material when subjected to elastic limit in a uniaxial stress state In uniaxial stress (yielding) ε > Hooke s law train energy density, U ij d ε ij > U U ε y d ε

24 train energy theory Body subjected to external loads > rincial stresses U train energy associated with rincial stresses ε ε ε ( ε ε ε ) υ υ υ ( ) ( ) ( ) U 4 [ υ( )] For onset of yielding, [ υ ( )]

25 train energy theory ield function f υ f ( ) ( ) quivalent stress > e υ ielding > f, safe f < e For D stress state > ield function becomes f υ For onset of yielding > f υ Plotting this in rincial stress sace 5

26 train energy theory Rearrange the terms η quivalent stress inside no failure ζ υ his reresents an ellise ransform to ζ-η csys 45 o ζ cos 45 η sin 45 ( ζ η) ζ sin 45 η cos 45 ubstitute these in the above exression ( ζ η) 6

27 train energy theory imlifying, ζ η ( υ) ( υ) emi major axis OA > ( υ ) emi minor axis OB > a b ( υ) ζ > a η b η B o A 45 o Higher Poisson ratio bigger major axis, smaller minor axis If υ > circle of radius ζ 7

28 train energy theory Pure shear Princial stresses corresonding to shear yield stress τ y τ y, -τ y τ y τ y ε ( υ), ε ( υ) train energy, U ( υ ) τ τ y > τ y.6 ( υ ) τ y 8

29 Distortional energy theory (von-mises) Hydrostatic loading alying uniform stress from all the directions on a body Large amount of strain energy can be stored xerimentally verified Pressures beyond yield Pressure alied from all sides stress no failure of material Hydrostatic loading change in size volume 9

30 von-mises theory nergy associated with volumetric change volumetric strain energy Volumetric strain energy no failure of material train energy causing material failure distortion energy associated with shear First invariant of deviatoric stress For a given stress state estimate distortion energy this should be less than distortion energy due to uniaxial tensile safe

31 von-mises theory Given stress state referred to rincial coordinate system [ ] ( ) ( ) ( ) ii J invariant, First > Princial strains > ε, ε, ε Volumetric strain > ε V ε ε ε

32 von-mises theory his gives ( ) ( ) { } ( ) ( ) ( ) υ υ ε υ ε ε ε ε V V ( ) ( ) ( ) ( ) ( ) ( ) & strains stresses rincial to due energy strain 6 energy, strain Volumetric υ υ υ υ ε U U U U V V V

33 von-mises theory Distortional energy m - m m m - m - m U D U - U V U D [( ) ( )] υ υ ( ) U D G imlifying this 6 [( ) ( ) ( ) ]

34 von-mises theory Comare this with distortion in uniaxial tensile stress U D 6G > G [( ) ( ) ( ) ] ( ) ( ) ( ) ield function, quivalent stress, e f [( ) ( ) ( ) ] e 4

35 von-mises theory Princial stresses of deviatoric shear stress, ii [ ] 5 ] [ ii ii > ii ii ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

36 von-mises theory imlifying this exression ( ) ( ) ( ) Hydrostatic ressure does not aear in the exression von-mises criteria has square terms result indeendent of signs of individual stress comonents Von-Mises equivalent stress > ve stress 6

37 von-mises theory D stress state > ield function, f Onset of yielding, Re-arrange the terms his reresents an ellise emi - major axis, OA emi - minor axis, OB η B o 45 o A ζ 7

38 von-mises theory Pure shear Princial stresses corresonding to shear yield stress τ y τ y, -τ y τ y > τ y. 577 hear yield.577 * ensile yield uitable for ductile materials 8

39 von-mises theory Plot yield function in D rincial stress sace n A Deviatoric lane B o f ( ) ( ) ( ) Cylinder, with hydrostatic stress as axis Axis makes equal DCs with all axes n i j k OA i j k 9

40 von-mises theory Projection of OA on hydrostatic axis >.. cos cos. k j i k j i OB n n OA OA OA n n OA θ θ 4 ( ) ( ) ( ) > k j i k j i OB k j i n OB OB OB OB OB OA BA BA OB OA BA r radius of cylinder

41 von-mises theory Radius of cylinder k j i R k j i k j i OB OA R BA 4 ( ) ( ) ( ) ( ) ( ) criteria ield tensor, stress deviatoris of First invariant R Radius J k j i R > > >

42 von-mises theory ield criteria, ( ) ( ) Use, 4 ( ) ( ) R R ielding deends on deviatoric stresses Hydrostatic stress has no role in yielding

43 von-mises theory econd invariant of deviatoric stress [ ] J J > 4 ( ) ( ) ( ) Redfining yield function J f J R R J J > > > J Materials

44 Max. shear stress theory (resca) ielding begins when the maximum shear stress at a oint equals the maximum shear stress at yield in a uniaxial tension τ max τ τ K > max K If maximum shear stress < / > No failure occurs For ure shear, τ y, -τ y τ max K > τ max τ y hear yield.5 ensile yield K τ y 44

45 resca theory In D stress state rincial stresses >, and Maximum shear stress,,. max 45 ield function,,. max yielding Onset of No yielding,,. max > > < f f K f

46 resca theory Following equations are obtained ( ) ( ) K f K f K K, ;, ± > 46 ( ) ( ) ( ) ( ) K f K f K K K f K f K K, ;,, ;, ± > ± >

47 resca theory Redefining yield function as, ( ) ( )( ) ( )( ) K K f,, ( ) ( ) ( ) ( ) ( ) ( ) 4,.,.,.,.,,, f f f f f f 47 ( ) ( )( ) ( )( ) ( )( ) K K K K K K f,, ach function reresents a lane in D rincial stress sace ( ) ( ) ( )( ) ( )( ) ( ) 4 4 4,, K K K f No effect of hydrostatic ressure in resca criteria

48 resca theory ield function in rincial stress sace A - K O - K Hydrostatic axis resca yield surface View A along hydrostatic axis 48

49 resca theory ield surface intersects rincial axes at K Wall/lane of hexagon - K B A θ O α C Hydrostatic axis Deviatoric lane, Hydrostati cos α α θ OB c axis 9 OA > > α > θ K OA cosθ K OB rojection of OA on deviatoric lane 49

50 resca theory resca hexagon K D A, B O C D OC OD cos - K O - K C > OC > OC K K 5

51 resca theory D stress state - ach equation reresents two lines in D stress sace ± K ± K ± K - K K C O K 45 K A - K K O 45 A K - K B - K - K ield curve elongated hexagon OB OA cos 45 OC B K OA cos 45 K K 5

52 resca theory D stress state - ach equation reresents two lines in D stress sace - K - K K - K - K O K 45 ield curve elongated hexagon C A K B - K ± ± K ± K ± K 5

53 von-mises resca theories Pure tension,, [( ) ( ) ( ) ] von-mises criteria > J K M 6 resca s criteria > K max,, K M, K Pure shear > τ y, -τ y > K M τ y K K M τ y, K τ y τ.577 (von Mises), τ.5 y y ( resca ) von-mises criteria redicts 5% higher shear stress than resca 5

54 von-mises resca theories D stress sace von-mises and resca η B o 45 o A ζ η B τ y o 45 o A ζ τ y τ y max. {,, } ielding in uniaxial tension ielding in shear resca conservative τ y τ y max.,, von-mises conservative 54

55 von-mises resca theories P xeriments by aylor & Quinney** hin walled tube subjected to axial and torsional loads xx A A τ xy xx P xx xx τ xy ** aylor and Quinney Plastic deformation of metals, Phil. rans. Roy. oc.a, -6, 9 xx xx τ xy 55

56 von-mises resca theories resca criteria xx τ xy > xx 4τ xy No yielding von-mises criteria if, xx τ xy / < No yielding if, xx xx τ xy τ / xy < 56

57 von-mises resca theories Plotting these two criteria τ xy /.6 resca Aluminium Mild steel Coer von-mises xx / xerimental data shows good agreement with von- Mises theory. resca conservative von-mises theory more accurate generally used in design xeriments show that for ductile materials yield in shear is.5 to.6 times of yield in tensile 57

58 Octahedral shear stress theory Octahedral lane makes equal angles with all rincial stress axes direction cosines same hear stress acting on this lane octahedral shear τ oct 9 [( ) ( ) ( ) ] Body subjected to ure tension,, [( ) ( ) ( ) ] τ oct 9 58

59 Octahedral shear stress theory Comaring this with von-mises theory > both are same Pure shear τ y, - τ y, τ y τ oct 9 6τ y [( ) ( ) ( ) ] [( ) ( ) ( ) ] τ oct τ y 6 τ y 9 ame as von-mises theory in ure shear Octahedral shear stress theory > von-mises theory 59

60 ensile & shear yield strengths ach failure theory gives a relation between yielding in tension and shear (υ.5) 6

61 Failure theories in a nut shell 6

62 6